What are the benefits and drawbacks of measuring inflation using index numbers such as CPI/RPI?

It's issue usually hinges on accuracy. Here's a few reasons, specifically around CPI:
Few households are average – the published figure for inflation is rarely the actual rate of inflation experienced by different people
1. The CPI is not fully representative - it will be inaccurate for the ‘non-typical’ household, e.g. 14% of the CPI index is devoted to motoring costs - inapplicable for non-car owners.
2. Spending patterns: e.g. Single people have different spending patterns from households that have one or more children
3. Changing quality of goods and services: Although the price of a good or service may rise, this may also be accompanied by improvements in quality / performance of the product
4. New products: The CPI is slow to respond to new products and services – the CPI basket is changed each year but only a few items fall out / come in